Sharp slope bounds for sweeping families of trigonal curves

We establish sharp bounds for the slopes of curves in Mg that sweep out the locus of trigonal curves, reproving Stankova-Frenkel's bound of 7 + 6/g for even g and obtaining the bound 7 + 20/(3g + 1) for odd g. For even g, we find an explicit expression of the so-called Maroni divisor in the Picard group of the space of admissible triple covers. For odd g, we describe the analogous extremal effective divisor and give a similar explicit expression.